This lecture defines the notion of cross-moment of a random vector, which is a generalization of the concept of moment of a random variable (see the lecture entitled Moments of a random variable).

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Let be a random vector. A cross-moment of is the expected value of the product of integer powers of the entries of :where is the -th entry of and are non-negative integers.

The following is a formal definition of cross-moment.

Definition
Let
be a
random vector. Let
and
.
Ifexists
and is finite, then it is called a **cross-moment** of
of order
.
If all cross-moments of order
exist and are finite, i.e. if (1) exists and is finite for all
-tuples
of non-negative integers
such that
,
then
is said to possess finite cross-moments of order
.

The following example shows how to compute a cross-moment of a discrete random vector.

Example Let be a discrete random vector and denote its components by , and . Let the support of be and its joint probability mass function beThe following is a cross-moment of of order :which can be computed by using the transformation theorem:

The central cross-moments of a random vector are just the cross-moments of the random vector of deviations .

Definition
Let
be a
random vector. Let
and
.
Ifexists
and is finite, then it is called a **central cross-moment** of
of order
.
If all central cross-moments of order
exist and are finite, that is, if (2) exists and is finite for all
-tuples
of non-negative integers
such that
,
then
is said to possess finite central cross-moments of order
.

The following example shows how to compute a central cross-moment of a discrete random vector.

Example Let be a discrete random vector and denote its components by , and . Let the support of beand its joint probability mass function beThe expected values of the three components of areThe following is a central cross-moment of of order :which can be computed by using the transformation theorem:

Please cite as:

Taboga, Marco (2021). "Cross-moments of a random vector", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-probability/cross-moments.

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